Successors of locally compact topological group topologies on abelian groups

Abstract

For a group $G$, let $\mathcal{G} (G)$ be the lattice of all topological group topologies on $G$. We prove that if $G$ is abelian, $\tau ,\sigma \in \mathcal{G} (G)$ and $\sigma $ is a successor of $\tau $ in $\mathcal{G} (G)$, then $\sigma $ is preco